ginac/inifcns.cpp

   1 /** @file inifcns.cpp
   2  *
   3  *  Implementation of GiNaC's initially known functions. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <vector>
  24 #include <stdexcept>
  25
  26 #include "inifcns.h"
  27 #include "ex.h"
  28 #include "constant.h"
  29 #include "lst.h"
  30 #include "matrix.h"
  31 #include "mul.h"
  32 #include "power.h"
  33 #include "operators.h"
  34 #include "relational.h"
  35 #include "pseries.h"
  36 #include "symbol.h"
  37 #include "symmetry.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 //////////
  43 // absolute value
  44 //////////
  45
  46 static ex abs_evalf(const ex & arg)
  47 {
  48         if (is_exactly_a<numeric>(arg))
  49                 return abs(ex_to<numeric>(arg));
  50
  51         return abs(arg).hold();
  52 }
  53
  54 static ex abs_eval(const ex & arg)
  55 {
  56         if (is_exactly_a<numeric>(arg))
  57                 return abs(ex_to<numeric>(arg));
  58         else
  59                 return abs(arg).hold();
  60 }
  61
  62 static void abs_print_latex(const ex & arg, const print_context & c)
  63 {
  64         c.s << "{|"; arg.print(c); c.s << "|}";
  65 }
  66
  67 static void abs_print_csrc_float(const ex & arg, const print_context & c)
  68 {
  69         c.s << "fabs("; arg.print(c); c.s << ")";
  70 }
  71
  72 REGISTER_FUNCTION(abs, eval_func(abs_eval).
  73                        evalf_func(abs_evalf).
  74                        print_func<print_latex>(abs_print_latex).
  75                        print_func<print_csrc_float>(abs_print_csrc_float).
  76                        print_func<print_csrc_double>(abs_print_csrc_float));
  77
  78
  79 //////////
  80 // Complex sign
  81 //////////
  82
  83 static ex csgn_evalf(const ex & arg)
  84 {
  85         if (is_exactly_a<numeric>(arg))
  86                 return csgn(ex_to<numeric>(arg));
  87
  88         return csgn(arg).hold();
  89 }
  90
  91 static ex csgn_eval(const ex & arg)
  92 {
  93         if (is_exactly_a<numeric>(arg))
  94                 return csgn(ex_to<numeric>(arg));
  95
  96         else if (is_exactly_a<mul>(arg) &&
  97                  is_exactly_a<numeric>(arg.op(arg.nops()-1))) {
  98                 numeric oc = ex_to<numeric>(arg.op(arg.nops()-1));
  99                 if (oc.is_real()) {
 100                         if (oc > 0)
 101                                 // csgn(42*x) -> csgn(x)
 102                                 return csgn(arg/oc).hold();
 103                         else
 104                                 // csgn(-42*x) -> -csgn(x)
 105                                 return -csgn(arg/oc).hold();
 106                 }
 107                 if (oc.real().is_zero()) {
 108                         if (oc.imag() > 0)
 109                                 // csgn(42*I*x) -> csgn(I*x)
 110                                 return csgn(I*arg/oc).hold();
 111                         else
 112                                 // csgn(-42*I*x) -> -csgn(I*x)
 113                                 return -csgn(I*arg/oc).hold();
 114                 }
 115         }
 116
 117         return csgn(arg).hold();
 118 }
 119
 120 static ex csgn_series(const ex & arg,
 121                       const relational & rel,
 122                       int order,
 123                       unsigned options)
 124 {
 125         const ex arg_pt = arg.subs(rel, subs_options::no_pattern);
 126         if (arg_pt.info(info_flags::numeric)
 127             && ex_to<numeric>(arg_pt).real().is_zero()
 128             && !(options & series_options::suppress_branchcut))
 129                 throw (std::domain_error("csgn_series(): on imaginary axis"));
 130
 131         epvector seq;
 132         seq.push_back(expair(csgn(arg_pt), _ex0));
 133         return pseries(rel,seq);
 134 }
 135
 136 REGISTER_FUNCTION(csgn, eval_func(csgn_eval).
 137                         evalf_func(csgn_evalf).
 138                         series_func(csgn_series));
 139
 140
 141 //////////
 142 // Eta function: eta(x,y) == log(x*y) - log(x) - log(y).
 143 // This function is closely related to the unwinding number K, sometimes found
 144 // in modern literature: K(z) == (z-log(exp(z)))/(2*Pi*I).
 145 //////////
 146
 147 static ex eta_evalf(const ex &x, const ex &y)
 148 {
 149         // It seems like we basically have to replicate the eval function here,
 150         // since the expression might not be fully evaluated yet.
 151         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 152                 return _ex0;
 153
 154         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 155                 const numeric nx = ex_to<numeric>(x);
 156                 const numeric ny = ex_to<numeric>(y);
 157                 const numeric nxy = ex_to<numeric>(x*y);
 158                 int cut = 0;
 159                 if (nx.is_real() && nx.is_negative())
 160                         cut -= 4;
 161                 if (ny.is_real() && ny.is_negative())
 162                         cut -= 4;
 163                 if (nxy.is_real() && nxy.is_negative())
 164                         cut += 4;
 165                 return evalf(I/4*Pi)*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 166                                       (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 167         }
 168
 169         return eta(x,y).hold();
 170 }
 171
 172 static ex eta_eval(const ex &x, const ex &y)
 173 {
 174         // trivial:  eta(x,c) -> 0  if c is real and positive
 175         if (x.info(info_flags::positive) || y.info(info_flags::positive))
 176                 return _ex0;
 177
 178         if (x.info(info_flags::numeric) &&      y.info(info_flags::numeric)) {
 179                 // don't call eta_evalf here because it would call Pi.evalf()!
 180                 const numeric nx = ex_to<numeric>(x);
 181                 const numeric ny = ex_to<numeric>(y);
 182                 const numeric nxy = ex_to<numeric>(x*y);
 183                 int cut = 0;
 184                 if (nx.is_real() && nx.is_negative())
 185                         cut -= 4;
 186                 if (ny.is_real() && ny.is_negative())
 187                         cut -= 4;
 188                 if (nxy.is_real() && nxy.is_negative())
 189                         cut += 4;
 190                 return (I/4)*Pi*((csgn(-imag(nx))+1)*(csgn(-imag(ny))+1)*(csgn(imag(nxy))+1)-
 191                                  (csgn(imag(nx))+1)*(csgn(imag(ny))+1)*(csgn(-imag(nxy))+1)+cut);
 192         }
 193
 194         return eta(x,y).hold();
 195 }
 196
 197 static ex eta_series(const ex & x, const ex & y,
 198                      const relational & rel,
 199                      int order,
 200                      unsigned options)
 201 {
 202         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 203         const ex y_pt = y.subs(rel, subs_options::no_pattern);
 204         if ((x_pt.info(info_flags::numeric) && x_pt.info(info_flags::negative)) ||
 205             (y_pt.info(info_flags::numeric) && y_pt.info(info_flags::negative)) ||
 206             ((x_pt*y_pt).info(info_flags::numeric) && (x_pt*y_pt).info(info_flags::negative)))
 207                         throw (std::domain_error("eta_series(): on discontinuity"));
 208         epvector seq;
 209         seq.push_back(expair(eta(x_pt,y_pt), _ex0));
 210         return pseries(rel,seq);
 211 }
 212
 213 REGISTER_FUNCTION(eta, eval_func(eta_eval).
 214                        evalf_func(eta_evalf).
 215                        series_func(eta_series).
 216                        latex_name("\\eta").
 217                        set_symmetry(sy_symm(0, 1)));
 218
 219
 220 //////////
 221 // dilogarithm
 222 //////////
 223
 224 static ex Li2_evalf(const ex & x)
 225 {
 226         if (is_exactly_a<numeric>(x))
 227                 return Li2(ex_to<numeric>(x));
 228
 229         return Li2(x).hold();
 230 }
 231
 232 static ex Li2_eval(const ex & x)
 233 {
 234         if (x.info(info_flags::numeric)) {
 235                 // Li2(0) -> 0
 236                 if (x.is_zero())
 237                         return _ex0;
 238                 // Li2(1) -> Pi^2/6
 239                 if (x.is_equal(_ex1))
 240                         return power(Pi,_ex2)/_ex6;
 241                 // Li2(1/2) -> Pi^2/12 - log(2)^2/2
 242                 if (x.is_equal(_ex1_2))
 243                         return power(Pi,_ex2)/_ex12 + power(log(_ex2),_ex2)*_ex_1_2;
 244                 // Li2(-1) -> -Pi^2/12
 245                 if (x.is_equal(_ex_1))
 246                         return -power(Pi,_ex2)/_ex12;
 247                 // Li2(I) -> -Pi^2/48+Catalan*I
 248                 if (x.is_equal(I))
 249                         return power(Pi,_ex2)/_ex_48 + Catalan*I;
 250                 // Li2(-I) -> -Pi^2/48-Catalan*I
 251                 if (x.is_equal(-I))
 252                         return power(Pi,_ex2)/_ex_48 - Catalan*I;
 253                 // Li2(float)
 254                 if (!x.info(info_flags::crational))
 255                         return Li2(ex_to<numeric>(x));
 256         }
 257
 258         return Li2(x).hold();
 259 }
 260
 261 static ex Li2_deriv(const ex & x, unsigned deriv_param)
 262 {
 263         GINAC_ASSERT(deriv_param==0);
 264
 265         // d/dx Li2(x) -> -log(1-x)/x
 266         return -log(_ex1-x)/x;
 267 }
 268
 269 static ex Li2_series(const ex &x, const relational &rel, int order, unsigned options)
 270 {
 271         const ex x_pt = x.subs(rel, subs_options::no_pattern);
 272         if (x_pt.info(info_flags::numeric)) {
 273                 // First special case: x==0 (derivatives have poles)
 274                 if (x_pt.is_zero()) {
 275                         // method:
 276                         // The problem is that in d/dx Li2(x==0) == -log(1-x)/x we cannot
 277                         // simply substitute x==0.  The limit, however, exists: it is 1.
 278                         // We also know all higher derivatives' limits:
 279                         // (d/dx)^n Li2(x) == n!/n^2.
 280                         // So the primitive series expansion is
 281                         // Li2(x==0) == x + x^2/4 + x^3/9 + ...
 282                         // and so on.
 283                         // We first construct such a primitive series expansion manually in
 284                         // a dummy symbol s and then insert the argument's series expansion
 285                         // for s.  Reexpanding the resulting series returns the desired
 286                         // result.
 287                         const symbol s;
 288                         ex ser;
 289                         // manually construct the primitive expansion
 290                         for (int i=1; i<order; ++i)
 291                                 ser += pow(s,i) / pow(numeric(i), _num2);
 292                         // substitute the argument's series expansion
 293                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 294                         // maybe that was terminating, so add a proper order term
 295                         epvector nseq;
 296                         nseq.push_back(expair(Order(_ex1), order));
 297                         ser += pseries(rel, nseq);
 298                         // reexpanding it will collapse the series again
 299                         return ser.series(rel, order);
 300                         // NB: Of course, this still does not allow us to compute anything
 301                         // like sin(Li2(x)).series(x==0,2), since then this code here is
 302                         // not reached and the derivative of sin(Li2(x)) doesn't allow the
 303                         // substitution x==0.  Probably limits *are* needed for the general
 304                         // cases.  In case L'Hospital's rule is implemented for limits and
 305                         // basic::series() takes care of this, this whole block is probably
 306                         // obsolete!
 307                 }
 308                 // second special case: x==1 (branch point)
 309                 if (x_pt.is_equal(_ex1)) {
 310                         // method:
 311                         // construct series manually in a dummy symbol s
 312                         const symbol s;
 313                         ex ser = zeta(_ex2);
 314                         // manually construct the primitive expansion
 315                         for (int i=1; i<order; ++i)
 316                                 ser += pow(1-s,i) * (numeric(1,i)*(I*Pi+log(s-1)) - numeric(1,i*i));
 317                         // substitute the argument's series expansion
 318                         ser = ser.subs(s==x.series(rel, order), subs_options::no_pattern);
 319                         // maybe that was terminating, so add a proper order term
 320                         epvector nseq;
 321                         nseq.push_back(expair(Order(_ex1), order));
 322                         ser += pseries(rel, nseq);
 323                         // reexpanding it will collapse the series again
 324                         return ser.series(rel, order);
 325                 }
 326                 // third special case: x real, >=1 (branch cut)
 327                 if (!(options & series_options::suppress_branchcut) &&
 328                         ex_to<numeric>(x_pt).is_real() && ex_to<numeric>(x_pt)>1) {
 329                         // method:
 330                         // This is the branch cut: assemble the primitive series manually
 331                         // and then add the corresponding complex step function.
 332                         const symbol &s = ex_to<symbol>(rel.lhs());
 333                         const ex point = rel.rhs();
 334                         const symbol foo;
 335                         epvector seq;
 336                         // zeroth order term:
 337                         seq.push_back(expair(Li2(x_pt), _ex0));
 338                         // compute the intermediate terms:
 339                         ex replarg = series(Li2(x), s==foo, order);
 340                         for (size_t i=1; i<replarg.nops()-1; ++i)
 341                                 seq.push_back(expair((replarg.op(i)/power(s-foo,i)).series(foo==point,1,options).op(0).subs(foo==s, subs_options::no_pattern),i));
 342                         // append an order term:
 343                         seq.push_back(expair(Order(_ex1), replarg.nops()-1));
 344                         return pseries(rel, seq);
 345                 }
 346         }
 347         // all other cases should be safe, by now:
 348         throw do_taylor();  // caught by function::series()
 349 }
 350
 351 REGISTER_FUNCTION(Li2, eval_func(Li2_eval).
 352                        evalf_func(Li2_evalf).
 353                        derivative_func(Li2_deriv).
 354                        series_func(Li2_series).
 355                        latex_name("\\mbox{Li}_2"));
 356
 357 //////////
 358 // trilogarithm
 359 //////////
 360
 361 static ex Li3_eval(const ex & x)
 362 {
 363         if (x.is_zero())
 364                 return x;
 365         return Li3(x).hold();
 366 }
 367
 368 REGISTER_FUNCTION(Li3, eval_func(Li3_eval).
 369                        latex_name("\\mbox{Li}_3"));
 370
 371 //////////
 372 // factorial
 373 //////////
 374
 375 static ex factorial_evalf(const ex & x)
 376 {
 377         return factorial(x).hold();
 378 }
 379
 380 static ex factorial_eval(const ex & x)
 381 {
 382         if (is_exactly_a<numeric>(x))
 383                 return factorial(ex_to<numeric>(x));
 384         else
 385                 return factorial(x).hold();
 386 }
 387
 388 REGISTER_FUNCTION(factorial, eval_func(factorial_eval).
 389                              evalf_func(factorial_evalf));
 390
 391 //////////
 392 // binomial
 393 //////////
 394
 395 static ex binomial_evalf(const ex & x, const ex & y)
 396 {
 397         return binomial(x, y).hold();
 398 }
 399
 400 static ex binomial_eval(const ex & x, const ex &y)
 401 {
 402         if (is_exactly_a<numeric>(x) && is_exactly_a<numeric>(y))
 403                 return binomial(ex_to<numeric>(x), ex_to<numeric>(y));
 404         else
 405                 return binomial(x, y).hold();
 406 }
 407
 408 REGISTER_FUNCTION(binomial, eval_func(binomial_eval).
 409                             evalf_func(binomial_evalf));
 410
 411 //////////
 412 // Order term function (for truncated power series)
 413 //////////
 414
 415 static ex Order_eval(const ex & x)
 416 {
 417         if (is_exactly_a<numeric>(x)) {
 418                 // O(c) -> O(1) or 0
 419                 if (!x.is_zero())
 420                         return Order(_ex1).hold();
 421                 else
 422                         return _ex0;
 423         } else if (is_exactly_a<mul>(x)) {
 424                 const mul &m = ex_to<mul>(x);
 425                 // O(c*expr) -> O(expr)
 426                 if (is_exactly_a<numeric>(m.op(m.nops() - 1)))
 427                         return Order(x / m.op(m.nops() - 1)).hold();
 428         }
 429         return Order(x).hold();
 430 }
 431
 432 static ex Order_series(const ex & x, const relational & r, int order, unsigned options)
 433 {
 434         // Just wrap the function into a pseries object
 435         epvector new_seq;
 436         GINAC_ASSERT(is_a<symbol>(r.lhs()));
 437         const symbol &s = ex_to<symbol>(r.lhs());
 438         new_seq.push_back(expair(Order(_ex1), numeric(std::min(x.ldegree(s), order))));
 439         return pseries(r, new_seq);
 440 }
 441
 442 // Differentiation is handled in function::derivative because of its special requirements
 443
 444 REGISTER_FUNCTION(Order, eval_func(Order_eval).
 445                          series_func(Order_series).
 446                          latex_name("\\mathcal{O}"));
 447
 448 //////////
 449 // Solve linear system
 450 //////////
 451
 452 ex lsolve(const ex &eqns, const ex &symbols, unsigned options)
 453 {
 454         // solve a system of linear equations
 455         if (eqns.info(info_flags::relation_equal)) {
 456                 if (!symbols.info(info_flags::symbol))
 457                         throw(std::invalid_argument("lsolve(): 2nd argument must be a symbol"));
 458                 const ex sol = lsolve(lst(eqns),lst(symbols));
 459
 460                 GINAC_ASSERT(sol.nops()==1);
 461                 GINAC_ASSERT(is_exactly_a<relational>(sol.op(0)));
 462
 463                 return sol.op(0).op(1); // return rhs of first solution
 464         }
 465
 466         // syntax checks
 467         if (!eqns.info(info_flags::list)) {
 468                 throw(std::invalid_argument("lsolve(): 1st argument must be a list"));
 469         }
 470         for (size_t i=0; i<eqns.nops(); i++) {
 471                 if (!eqns.op(i).info(info_flags::relation_equal)) {
 472                         throw(std::invalid_argument("lsolve(): 1st argument must be a list of equations"));
 473                 }
 474         }
 475         if (!symbols.info(info_flags::list)) {
 476                 throw(std::invalid_argument("lsolve(): 2nd argument must be a list"));
 477         }
 478         for (size_t i=0; i<symbols.nops(); i++) {
 479                 if (!symbols.op(i).info(info_flags::symbol)) {
 480                         throw(std::invalid_argument("lsolve(): 2nd argument must be a list of symbols"));
 481                 }
 482         }
 483
 484         // build matrix from equation system
 485         matrix sys(eqns.nops(),symbols.nops());
 486         matrix rhs(eqns.nops(),1);
 487         matrix vars(symbols.nops(),1);
 488
 489         for (size_t r=0; r<eqns.nops(); r++) {
 490                 const ex eq = eqns.op(r).op(0)-eqns.op(r).op(1); // lhs-rhs==0
 491                 ex linpart = eq;
 492                 for (size_t c=0; c<symbols.nops(); c++) {
 493                         const ex co = eq.coeff(ex_to<symbol>(symbols.op(c)),1);
 494                         linpart -= co*symbols.op(c);
 495                         sys(r,c) = co;
 496                 }
 497                 linpart = linpart.expand();
 498                 rhs(r,0) = -linpart;
 499         }
 500
 501         // test if system is linear and fill vars matrix
 502         for (size_t i=0; i<symbols.nops(); i++) {
 503                 vars(i,0) = symbols.op(i);
 504                 if (sys.has(symbols.op(i)))
 505                         throw(std::logic_error("lsolve: system is not linear"));
 506                 if (rhs.has(symbols.op(i)))
 507                         throw(std::logic_error("lsolve: system is not linear"));
 508         }
 509
 510         matrix solution;
 511         try {
 512                 solution = sys.solve(vars,rhs,options);
 513         } catch (const std::runtime_error & e) {
 514                 // Probably singular matrix or otherwise overdetermined system:
 515                 // It is consistent to return an empty list
 516                 return lst();
 517         }
 518         GINAC_ASSERT(solution.cols()==1);
 519         GINAC_ASSERT(solution.rows()==symbols.nops());
 520
 521         // return list of equations of the form lst(var1==sol1,var2==sol2,...)
 522         lst sollist;
 523         for (size_t i=0; i<symbols.nops(); i++)
 524                 sollist.append(symbols.op(i)==solution(i,0));
 525
 526         return sollist;
 527 }
 528
 529 /* Force inclusion of functions from inifcns_gamma and inifcns_zeta
 530  * for static lib (so ginsh will see them). */
 531 unsigned force_include_tgamma = tgamma_SERIAL::serial;
 532 unsigned force_include_zeta1 = zeta1_SERIAL::serial;
 533
 534 } // namespace GiNaC